On Equivalence Between Network Topologies

TL;DR

This paper introduces a hierarchical method to approximate the capacity of complex multi-source networks by simplifying their structure, providing bounds that are computationally feasible and sometimes exact.

Contribution

A novel hierarchical approach for bounding network capacity by replacing components with simpler structures, improving computational tractability.

Findings

The method provides bounds with quantifiable accuracy.

Some network families can be simplified without loss of capacity accuracy.

Abstract

One major open problem in network coding is to characterize the capacity region of a general multi-source multi-demand network. There are some existing computational tools for bounding the capacity of general networks, but their computational complexity grows very quickly with the size of the network. This motivates us to propose a new hierarchical approach which finds upper and lower bounding networks of smaller size for a given network. This approach sequentially replaces components of the network with simpler structures, i.e., with fewer links or nodes, so that the resulting network is more amenable to computational analysis and its capacity provides an upper or lower bound on the capacity of the original network. The accuracy of the resulting bounds can be bounded as a function of the link capacities. Surprisingly, we are able to simplify some families of network structures without any loss in accuracy.